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Math  /  Calculus

QuestionFind the equation of the tangent line at the given point on the curve. x−sin⁡(πy)=5;(5,4)x-\sin (\pi y)=5 ;(5,4)

Studdy Solution

STEP 1

1. The curve is given by the equation x−sin⁡(πy)=5 x - \sin(\pi y) = 5 .
2. We need to find the equation of the tangent line at the point (5,4) (5, 4) .
3. The tangent line can be found using the derivative of the curve.

STEP 2

1. Differentiate the given equation implicitly with respect to x x .
2. Evaluate the derivative at the given point to find the slope of the tangent line.
3. Use the point-slope form to find the equation of the tangent line.

STEP 3

Differentiate the given equation implicitly with respect to x x . The given equation is:
x−sin⁡(πy)=5 x - \sin(\pi y) = 5
Differentiating both sides with respect to x x , we get:
ddx(x)−ddx(sin⁡(πy))=ddx(5) \frac{d}{dx}(x) - \frac{d}{dx}(\sin(\pi y)) = \frac{d}{dx}(5)
The derivative of x x with respect to x x is 1 1 , and the derivative of a constant is 0 0 . For the term −sin⁡(πy)-\sin(\pi y), we use the chain rule:
1−cos⁡(πy)⋅π⋅dydx=0 1 - \cos(\pi y) \cdot \pi \cdot \frac{dy}{dx} = 0

STEP 4

Rearrange the equation to solve for dydx\frac{dy}{dx}:
1=πcos⁡(πy)⋅dydx 1 = \pi \cos(\pi y) \cdot \frac{dy}{dx}
dydx=1πcos⁡(πy) \frac{dy}{dx} = \frac{1}{\pi \cos(\pi y)}

STEP 5

Evaluate the derivative at the given point (5,4) (5, 4) . Substitute y=4 y = 4 into the derivative:
dydx=1πcos⁡(π⋅4) \frac{dy}{dx} = \frac{1}{\pi \cos(\pi \cdot 4)}
Since cos⁡(4π)=1 \cos(4\pi) = 1 , we have:
dydx=1π⋅1=1π \frac{dy}{dx} = \frac{1}{\pi \cdot 1} = \frac{1}{\pi}

STEP 6

Use the point-slope form of the equation of a line to find the equation of the tangent line. The point-slope form is:
y−y1=m(x−x1) y - y_1 = m(x - x_1)
where m m is the slope and (x1,y1) (x_1, y_1) is the point. Substitute m=1π m = \frac{1}{\pi} and (x1,y1)=(5,4) (x_1, y_1) = (5, 4) :
y−4=1π(x−5) y - 4 = \frac{1}{\pi}(x - 5)
The equation of the tangent line is:
y=1πx−5π+4 y = \frac{1}{\pi}x - \frac{5}{\pi} + 4

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